I teach a course in real analysis and applications to partial differential equations in which I spend some weeks talking about Sobolev spaces. I have always used the symbol $C_0^\infty(\Omega)$ to denote the set of infinitely differentiable functions on the (open) subset $\Omega$ of ${\Bbb R}^n$ with compact support. A colleague once told me that this notation is bad and misleading, so that I should switch to $C_c^\infty(\Omega)$.
I admit that I could not understand his remark. After thiking about it, the only idea that came to my mind is that somebody might use the first notation to denote the set of those functions $u \colon \Omega \to \mathbb{R}^n$ that vanish at infinity (provided that $\Omega$ "contains" infinity). For example, when $\Omega$ is the whole space, we request that for every $\varepsilon>0$ there exists a compact subset $K$ such that $\sup_{x \in \complement K} |u(x)|<\varepsilon$.
My question is: is there any other reason why I should stop using $C_0^\infty$?